Which one of the following options is correct for the given data in the table?

Iteration(i)	0	1	2	3
Input(I)	20	-4	10	15
Output(X)	20	16	26	41
Output(Y)	20	-80	-800	-12000

The correct option is:
X(i) = X(i − 1) + I(i)
Y(i) = Y(i − 1) × I(i), i > 0

To verify which mathematical relation correctly represents the given data, we can test the recurrence relations for each iteration step (where
$i>0$
).

Let us analyze the values for Output(X) and Output(Y) step-by-step using the correct relations:

1. Verification of the relation for Output(X):
The proposed relation is:

$X\left(i\right)=X(i-1)+I\left(i\right)$

Let us check this relation for iterations
$i=1,2,3$
:

• For iteration
$i=1$
:

$X\left(1\right)=X\left(0\right)+I\left(1\right)=20+(-4)=16$

This matches the table value of Output(X) at iteration 1, which is 16.

• For iteration
$i=2$
:

$X\left(2\right)=X\left(1\right)+I\left(2\right)=16+10=26$

This matches the table value of Output(X) at iteration 2, which is 26.

• For iteration
$i=3$
:

$X\left(3\right)=X\left(2\right)+I\left(3\right)=26+15=41$

This matches the table value of Output(X) at iteration 3, which is 41.

2. Verification of the relation for Output(Y):
The proposed relation is:

$Y\left(i\right)=Y(i-1)\times I\left(i\right)$

Let us check this relation for iterations
$i=1,2,3$
:

• For iteration
$i=1$
:

$Y\left(1\right)=Y\left(0\right)\times I\left(1\right)=20\times (-4)=-80$

This matches the table value of Output(Y) at iteration 1, which is -80.

• For iteration
$i=2$
:

$Y\left(2\right)=Y\left(1\right)\times I\left(2\right)=-80\times 10=-800$

This matches the table value of Output(Y) at iteration 2, which is -800.

• For iteration
$i=3$
:

$Y\left(3\right)=Y\left(2\right)\times I\left(3\right)=-800\times 15=-12000$

This matches the table value of Output(Y) at iteration 3, which is -12000.

Since both Output(X) and Output(Y) values calculated step-by-step match the given table data perfectly, the relation is verified as correct.